Use the free Square Root Calculator below: your data never leaves your browser — all work happens on your device, and nothing is uploaded to any server.
What This Calculator Does
A square root calculator finds the number that, multiplied by itself, gives your number. The square root of 144 is 12, because 12 times 12 is 144. The square root of 2 is about 1.414, because 1.414 times 1.414 is about 2. Square roots undo squaring, just as subtraction undoes addition. They appear everywhere: in the Pythagorean theorem for right triangles, in statistics for standard deviation, in geometry for areas, and in everyday problems like finding the side of a square garden from its area.
This tool gives you more than a decimal. It tells you whether your number is a perfect square, meaning its root is a whole number, and if not, it shows the two nearest perfect squares that bracket it. For example, 50 is not a perfect square, but it sits between 49 (7 squared) and 64 (8 squared), so its root sits between 7 and 8. The tool also checks its answer by multiplying the root by itself. For negative numbers it explains honestly that there is no real square root, and shows the imaginary root instead.
How to Use It
Type any number and click Calculate. Positive numbers give a real root. Zero gives zero. Negative numbers give an explanation plus the imaginary root: the square root of negative 4 is 2i, where i is the imaginary unit, defined as the square root of negative 1. If you are working on school homework, check whether your problem expects real roots only. Most school problems do, and a negative number under a square root usually means you should re-check an earlier step.
Perfect Squares and Estimation, Explained
Perfect squares are the squares of whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. Knowing a few by heart makes estimation fast. The square root of 50 must be just above 7, because 50 is just above 49. The square root of 200 must be just above 14, because 14 squared is 196. The tool uses this bracketing idea to describe non-perfect squares. It also matters for checking answers: if you compute a root of 50 and get 25, something is wrong, because 25 squared is 625, far above 50.
A Worked Example
Suppose a square garden has an area of 144 square metres. Its side is the square root of 144, which is 12 metres, because 12 times 12 is 144. Now suppose the area is 200 square metres. The root is about 14.142 metres, because 14.142 times 14.142 is about 200. If you buy fencing for that garden, you need four sides, so about 56.6 metres of fencing. A third example: the square root of 2 is about 1.414. That number is famous because it is the diagonal of a 1-by-1 square, and it cannot be written as an exact fraction.
Common Mistakes
The most common mistake is forgetting that every positive number has two square roots: a positive one and a negative one. The square roots of 144 are 12 and negative 12, because negative 12 times negative 12 is also 144. Calculators show the principal, positive root by convention. Another mistake is distributing a root over addition: the square root of 9 plus 16 is 5, not 3 plus 4, because you must add first (25) and then take the root. A third mistake is rounding too early in multi-step problems; keep extra digits until the final step.
Frequently Asked Questions
What is the square root of 0? Zero, because zero times zero is zero. What is the square root of 1? One, and also negative one. Why do negative numbers have no real root? Because any real number multiplied by itself is zero or positive. A negative times a negative is positive, so no real number squares to a negative. Is the square root of 2 rational? No. It is irrational, with a decimal that never repeats or ends.